Beating the Standard Quantum Limit under Ambient Conditions with Solid-State Spins 1,2,3 1,2,3 4 1,2,3 Tianyu Xie †, Zhiyuan Zhao †, Xi Kong , Wenchao Ma ‡ Mengqi Wang1,2,3, Xiangyu Ye2, Pei Yu2, Zhiping Yang2, Shaoyi Xu2 1,2,3 2,3 1,2,3 1,2,3 Pengfei Wang , Ya Wang , Fazhan Shi ∗ & Jiangfeng Du ∗ 1Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China 2CAS Key Laboratory of Microscale Magnetic Resonance and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China 3Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China 4National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China †These authors contributed equally to this work. Present address: Department of Chemistry, Massachusetts Institute of Technology, ‡ Cambridge, Massachusetts 02139, USA ∗E-mail: [email protected] ∗E-mail: [email protected] arXiv:2101.12048v1 [quant-ph] 28 Jan 2021 Precision measurement plays a crucial role in all fields of science. The use of entangled sensors in quantum metrology improves the precision limit from the standard quantum limit (SQL) to the Heisenberg limit (HL) [1,2]. To date, most experiments beating the SQL are performed on the sensors which are well isolated under extreme conditions [3–13]. However, it has not been realized in solid-state spin systems at ambient conditions, owing to its intrinsic complexity for the preparation and survival of pure and entangled quan- 1 tum states. Here we show a full interferometer sequence beating the SQL by employing a hybrid multi-spin system, namely the nitrogen-vacancy (NV) defect in diamond. The interferometer sequence starts from a deterministic and joint initialization, undergoes entanglement and disentanglement of multiple spins, and ends up with projective mea- surement. In particular, the deterministic and joint initialization of NV negative state, NV electron spin, and two nuclear spins is realized at room temperature for the first time. By means of optimal control, non-local gates are implemented with an estimated fidelity above the threshold for fault-tolerant quantum computation. With these techniques com- bined, we achieve two-spin interference with a phase sensitivity of 1.79 ± 0.06 dB beyond the SQL and three-spin 2.77 ± 0.10 dB. Moreover, the deviations from the HL induced by experimental imperfections are completely accountable. The techniques used here are of fundamental importance for quantum sensing and computing [14, 15], and naturally applicable to other solid-state spin systems [16–19]. Over the last decades, much effort and progress has been made to realize sub-SQL mea- surements in many systems such as trapped ions [3,4], cold thermal atoms [5–7], Bose-Einstein condensates (BEC) [8–10], photons [11, 12], and mechanical systems [13]. These systems are appropriate for some extraordinary applications including precision tests of the fundamental laws of physics [20], atomic clocks [8,10] and gravitational wave detection [11]. However, their sensors only work well under extreme conditions, and thus not suitable for most realistic mea- surements demanding moderate conditions or a high spatial resolution, e.g., microscale imag- ing at ambient conditions. Fortunately, a recently emergent solid-state spin system, nitrogen- vacancy centers in diamond, provides such an approach. With this atomic-scale sensor, electron and nuclear magnetic resonance of single molecules have been accomplished [21, 22], while nanoscale magnetic imaging combined with an atomic force microscope is being put into prac- tice [23, 24]. Though there are some NV-based work related to the SQL [25–27], the canonical 2 method of using entanglement to beat the SQL remains elusive. The main challenge lies in the difficulty of deterministically initializing solid-state spins to a pure state and deeply entangling the spins at ambient conditions. Here we demonstrate the entire measurement process for an entangled interferometer based on a single NV defect center. The NV center in diamond is formed by a substitutional nitrogen atom and an adjacent vacancy as shown in Fig. 1a. The NV electron spin (S = 1), the attendant 14N nuclear spin (I = 1) and one of randomly distributed 13C nuclear spins (I = 1/2) constitute the interferometer. Through hyperfine interactions between the electron spin and nuclear spins, multi-qubit quantum entanglement is producible. Complicated dynamics involving multiple degrees of freedom (charges, orbits and spin angular momenta) are driven by 532-nm green laser and 594-nm orange laser for the initialization and readout of the NV charge state and the electron spin detailed in Fig. 1b. Microwave (MW) and radio-frequency (RF) pulses imposed by a coplanar waveguide can coherently manipulate the electron spin and two nuclear spins both locally and non-locally (Fig. 1c). Projective measurement is an indispensable ingredient to beat the SQL. It is implemented via the sequence of the inset of Fig. 1d for a nuclear spin [28, 29]. The sequence eliminates the classical noise via thousands of readouts of the nuclear spin whose information is correlated with the electron spin via a CnNOTe (an electron NOT gate conditional on nuclear state) before the electron spin readout during every cycle. The non-destructive nature of the nuclear spin after every readout is brought about by a high magnetic field (8066 G in our setup). The result of implementing the sequence is a two-peak statistic of photon counts for the 13C nuclear spin (Fig. 1d). We obtain a readout fidelity of 99.14(4)% by setting an optimal threshold to determine the spin state. Deterministic initialization to a pure enough state is equally crucial. Especially, it is rather difficult for solid-state spin system at room temperature resulting from the complicated dynam- 3 ics of the local defect electron (Fig. 1b) exposed to a thermal phonon sea and energy bands, unlike well isolated ions trapped in electromagnetic field [3, 4]. Existing methods can proba- bilistically initialize NV negative state and nuclear spins heralded by measurement outcomes with abandoning a considerable proportion of data, which will dramatically reduce the sensi- tivity [15]. Here we propose a brand new scheme to deterministically and jointly initialize NV charge state, electron spin and two nuclear spins with a high fidelity, in combination with two novel techniques developed recently [30,31]. The overall sequence, depicted in Fig. 2a, initial- izes our system to the joint state NV−, m = 0 m = 1/2 m = +1 with a fidelity | S i ⊗ | C − i ⊗ | N i of 93.3(3)% (see Supplementary Section 4.4). The scheme includes three elaborately devised techniques, i.e., real-time feedback for NV negative state preparation (Fig. 2b) [30], chopped laser sequence for a better spin polarization [31, 32] without destroying charge state (Fig. 2c) and exquisite optimal control and population shelving for polarization transfer to nuclear spins (Fig. 2d). As shown in Fig. 1b, 594-nm orange laser excites the NV negative state but does not excite the neutral state. Therefore, the negative state is prepared if one photon is collected during laser illumination, and if not do it again. Before photon counting during every cycle, the green laser is switched on to mix charge states (Fig. 2b). The fidelity is estimated to be 98.94(3)% by single-shot readout of charge state (see Supplementary Section 4.1). Conventionally, a squared pulse of green laser is applied for polarizing the electron spin to m = 0 with a fidelity of 90.0(3)%. Here it is replaced by | S i a chopped laser sequence plotted in Fig. 2c with an improved fidelity of 97.8(2) % (see Sup- plementary Section 4.2). Most strikingly, the NV negative state can survive through it with a near-unit probability of 99.42(5)% obtained from the lower part of Fig. 2c. With these two tech- niques combined, the defect is initialized in NV−, m = 0 . After that, polarization transfer | S i from the electron spin via swap-like gates initializes the nuclear spins with the estimated fideli- ties of 98.34(13)% for 13C and 98.71(18)% for 14N (see Supplementary Section 4.3). In order 4 to reduce the polarization loss induced by imperfection of non-local operation, the technique of optimal control is harnessed to achieve robust and high-fidelity state manipulation with one sequence displayed in Fig. 2d. To further diminish the residual error, the population of the state flipped by C NOT is depleted by shelving to m = +1 indicated by the dashed box of Fig. n e | S i 2a. The techniques aforementioned can be compatibly assembled, mainly benefitting from two vast gaps: one between the green laser duration 4 µs in real-time feedback and the lifetimes of ∼ nuclear spins under laser illumination 40 ms for 13C and 10 ms for 14N; the other between ∼ ∼ the laser duration 2.5 µs in chopped laser sequence and the lifetime of NV negative state 0.4 ∼ ∼ ms illuminated by chopped laser sequence. The main sequence performing the entanglement-based interference is displayed in Fig. 3a. Here we prepared the Greenberger-Horne-Zeilinger (GHZ) state that is maximally metrologi- cally useful. In comparison to directly measuring the parity of the GHZ state after acquiring an overall phase Nφ [4], we disentangle the state and the phase Nφ enters into only one of the spins with other spins separable. In this way, the projective measurement of only one spin is needed, leading to the reduction of the readout error (the readout fidelity for 14N is only 97.02(6)%). To entangle nuclear spins, we adopted a novel scheme used in [15] to realize a conditional π-phase (CPhase) gate on the electron spin.